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Stability of a boundary spike layer for the Gierer–Meinhardt system

Published online by Cambridge University Press:  09 November 2005

YASUHITO MIYAMOTO
Affiliation:
Meme Media Laboratory, Hokkaido University: Kita 13 Nishi 8, Kita-ku, Sapporo 060-0813, Japan e-mail: miyamoto@nsc.es.hokudai.ac.jp

Abstract

We consider the activator-inhibitor Gierer–Meinhardt reaction-diffusion system of biological pattern formation in a closed bounded domain. The existence and stability of a boundary apike-layer solution to the Gierer–Meinhardt model, and it, so-called shadow limit, is analysed. In the limit of small activator diffusivity, together with a large inhibitor diffusivity, an equilibrium boundary spike-layer solution is constructed that concentrates at a non-degenerate critical point P of the boundary. By non-degenerate we mean that every principal curvature of the boundary has a local maximum at P, and hence the mean curvature at the boundary has a local maximum at P. Rigorous results for the stability of such a boundary spike-layer solution are given.

Type
Papers
Copyright
2005 Cambridge University Press

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